Connected Green functions and their one-particle irreducible components

A. L. Kuzemsky

Lukong Cornelius Fai

This chapter explores connected Green's functions with one-particle and two-particle irreducible vertices. It allows reformulating and simplifying of the perturbation expansion. One cornerstone of the functional techniques is the concept of generating functionals that are obtained by coupling the field operators to one or more external fields, called the source fields. These source fields are thought of as our probes inside the quantum system. The introduction of the source term is artificial and serves for technical reasons because it permits to write higher-order correlation functions via derivatives. Once the derivative is taken, we set the source to zero. However, there are existing systems with source terms having a physical meaning, such as open systems. The chapter focuses on the full 2n-point correlation function generated via a source field coupled to fermion operators. One-particle irreducible Green's functions play an important role in the renormalization of quantum field theories in addition to nonperturbative calculations.

Tudor D. Stanescu, Gabriel Kotliar

We investigate the low-energy properties of a correlated metal in the proximity of a Mott insulator within the Hubbard model in two dimensions. We introduce a version of the cellular dynamical mean-field theory using cumulants as the basic irreducible objects. The cumulants are used for reconstructing the lattice quantities from their cluster counterparts. The zero-temperature one-particle Green function is characterized by the appearance of lines of zeros, in addition to a Fermi surface which changes topology as a function of doping. We show that these features are intimately connected to the opening of a pseudogap in the one-particle spectrum and provide a simple picture for the appearance of Fermi arcs.

Nikolai Borisovich Melnikov, Boris Il'ich Reser

A. A. Katanin

We consider formulations of the functional renormalization-group (fRG) flow for correlated electronic systems with the dynamical mean-field theory as a starting point. We classify the corresponding renormalization-group schemes into those neglecting one-particle irreducible six-point vertices (with respect to the local Green’s functions) and neglecting one-particle reducible six-point vertices. The former class is represented by the recently introduced DMF 2 RG approach [31], but also by the scale-dependent generalization of the one-particle irreducible representation (with respect to local Green’s functions, 1PI-LGF) of the generating functional [20]. The second class is represented by the fRG flow within the dual fermion approach [16, 32]. We compare formulations of the fRG approach in each of these cases and suggest their further application to study 2D systems within the Hubbard model.